It is straightforward to verify that Property \((i)\)  implies Property \((ii)\)  (the seminorm property holds with \(s=2\)  and one can use appropriate rotations on \(\ell \)  to verify the equidistribution property).
So we only prove that Property \((ii)\)  implies Property \((i)\) .
Let \(\mu =\int \mu _x\, d\mu \)  be the ergodic decomposition of the measure \(\mu \) . Since
\({\mathbb {E}}(f|\mathcal {K}_{rat}(\mu ))=0\)  implies that \({\mathbb {E}}(f|\mathcal {K}_{rat}(\mu _x))=0\)  for \(\mu _x\)  almost every \(x\in X\)  (see for example [1]}), we can assume that the system \((X,\mu ,T)\)  is ergodic.
We claim that it suffices to show that the Kronecker factor \(\mathcal {K}(T)\)  is characteristic for mean convergence of the averages
\({\mathbb {E}}_{n\in [N]}\, T^{a_1(n)}f_1 \cdot \ldots \cdot T^{a_\ell (n)}f_\ell ,\) 
meaning that the previous averages converge to 0 in \(L^2(\mu )\)  as \(N\rightarrow \infty \)  if at least one of the functions is orthogonal to \(\mathcal {K}(T)\)  (recall that \(\mathcal {K}(T)\)  is \(L^2(\mu )\) -closure of the linear span of the eigenfunctions of the system).
Indeed, if this is the case, then by approximation, we can assume that all functions are eigenfunctions; hence, for \(i=1,\ldots , \ell \)  we have
\(T^{a_i(n)}f_i=e(a_i(n)t_i)\, f_i, \quad n\in {\mathbb {N}},\) 
for some \(t_i\in [0,1)\) , not all of them rational (since at least one of the functions is orthogonal to \(\mathcal {K}_{rat}(T)\) ). In this case, the needed convergence to zero follows from
our assumption that the sequences \(a_1,\ldots , a_\ell \)  are good for irrational equidistribution.
So it remains to show that if \({\mathbb {E}}(f_i|\mathcal {K}(T))=0\) , for some \(i\in \lbrace 1,\ldots , \ell \rbrace \) , then the averages (REF ) converge to 0 in \(L^2(\mu )\) . Without loss of generality we can assume that \(i=1\) , hence \({\mathbb {E}}(f_1|\mathcal {K}(T))=0\) .
Since the sequences \(a_1,\ldots , a_\ell \)  are very good for seminorm estimates and \({\mathbb {E}}(f|\mathcal {Z}_s(T))=0\)  implies \(\vert \!|\!| f|\!|\!\vert _{s+1,T}=0\) , we can assume that all functions \(f_1,\ldots , f_\ell \)  are \(\mathcal {Z}_s(T)\) -measurable for some \(s\in {\mathbb {N}}\) . In this case we can assume that \(\mathcal {X}=\mathcal {Z}_s(T)\) ,The assumption that the sequences \(a_1,\ldots , a_\ell \)  are very good for seminorm estimates (versus simply “good”) is needed in order to get this reduction. hence using Theorem REF  and an approximation argument we can assume that the system \((X,\mu ,T)\)  has finite rational spectrum, in which case there exists \(k\in {\mathbb {N}}\)  such the system \((X,\mu ,T^k)\)  has trivial rational spectrum.
For \(i=1,\ldots , \ell \)  we let
\(b_i(n):=\sum _{r=0}^{k-1} {\bf 1}_{k{\mathbb {Z}}+r}(a_i(n)) \, \frac{a_i(n)-r}{k}, \quad n\in {\mathbb {Z}}.\) 
By Lemma REF , the sequences \(b_1,\ldots , b_\ell \)  are good for irrational equidistribution and seminorm estimates.
Then for \(i=1,\ldots , \ell \)  and \(S:=T^k\)  we have that
\(T^{a_i(n)}f_i=\sum _{r=0}^{k-1} {\bf 1}_{k{\mathbb {Z}}+r}(a_i(n))\, (T^k)^{\frac{a_i(n)-r}{k}}(T^rf_i)=\sum _{r=0}^{k-1} {\bf 1}_{k{\mathbb {Z}}+r}(a_i(n))\, S^{b_i(n)}(T^rf_i), \quad n\in {\mathbb {N}}.\) 
We insert this identity in (REF ) and expand the product.
We deduce
that it suffices to show the following: If the (not necessarily ergodic) system \((X,\mu ,S)\)  has trivial rational spectrum, the sequences \(b_1,\ldots , b_\ell \colon {\mathbb {N}}\rightarrow {\mathbb {Z}}\)  are good for irrational equidistibution and seminorm estimates,
and \({\mathbb {E}}(f_1|\mathcal {K}(T))=0\) , then for every bounded sequence \((c_n)\)  we have
\(\lim _{N\rightarrow \infty }{\mathbb {E}}_{n\in [N]}\, c_n\cdot S^{b_1(n)}f_1 \cdot \ldots \cdot S^{b_\ell (n)}f_\ell =0\) 
in \(L^2(\mu )\) .
To prove this, we first remark that since the system \((X,\mu ,S)\)  has trivial rational spectrum, the same holds for the system
\((X\times X,\mu \times \mu ,S\times S)\) . Furthermore, it is known that
\({\mathbb {E}}(f_1|\mathcal {K}(S))=0\)  implies that
\({\mathbb {E}}(f_1\otimes \overline{f}_1|\mathcal {I}(S\times S))=0\) . Combining these facts (whose proof follows for example from [2]}) we deduce that if
\(\mu \times \mu =\int (\mu \times \mu )_{(x,y)}\, d(\mu \times \mu )\) 
is the ergodic decomposition of the measure \(\mu \times \mu \)  with respect to the transformation \(S\times S\) , then
for \((\mu \times \mu )\) -almost every \((x,y)\in X\times X\)  the system \((X\times X,(\mu \times \mu )_{(x,y)},S\times S)\)  is ergodic, has trivial rational spectrum, and \(\int f_1\otimes \overline{f}_1\, d(\mu \times \mu )_{(x,y)}=0\) . By
Theorem REF  we deduce that for \((\mu \times \mu )\) -almost every \((x,y)\in X\times X\)  we have
\({\mathbb {E}}_{n\in [N]}\, (S\times S)^{b_1(n)}(f_1\otimes \overline{f}_1) \cdot \ldots \cdot (S\times S)^{b_\ell (n)}(f_\ell \otimes \overline{f}_\ell )\rightarrow ^{L^2((\mu \times \mu )_{(x,y)})}0\) 
as \(N\rightarrow \infty \) .
This implies that
\({\mathbb {E}}_{n\in [N]}\, (S\times S)^{b_1(n)}(f_1\otimes \overline{f}_1) \cdot \ldots \cdot (S\times S)^{b_\ell (n)}(f_\ell \otimes \overline{f}_\ell )\rightarrow ^{L^2(\mu \times \mu )}0\) 
as \(N\rightarrow \infty \) ,
and using the Cauchy-Schwarz inequality we deduce that
\(\lim _{N\rightarrow \infty }{\mathbb {E}}_{n\in [N]}\, \int (f_{0,N}\otimes \overline{f}_{0,N})\cdot (S\times S)^{b_1(n)}(f_1\otimes \overline{f}_1) \cdot \ldots \cdot (S\times S)^{b_\ell (n)}(f_\ell \otimes \overline{f}_\ell )\, d(\mu \times \mu )=0\) 
for all functions \(f_{0,N}\in L^\infty (\mu )\) , \(N\in {\mathbb {N}}\) , that are uniformly bounded.
Hence,
\(\lim _{N\rightarrow \infty } {\mathbb {E}}_{n\in [N]}\, \Big |\int f_{0,N} \cdot S^{b_1(n)}f_1 \cdot \ldots \cdot S^{b_\ell (n)}f_\ell \, d\mu \Big |^2= 0,\) 
which implies using the Cauchy-Schwarz inequality that
\(\lim _{N\rightarrow \infty } {\mathbb {E}}_{n\in [N]}\,c_n\, \int f_{0,N} \cdot S^{b_1(n)}f_1 \cdot \ldots \cdot S^{b_\ell (n)}f_\ell \, d\mu = 0\) 
for every bounded sequence \((c_n)\) .
If we let
\(f_{0,N}:= \overline{{\mathbb {E}}_{n\in [N]}\, c_n \cdot S^{b_1(n)}f_1 \cdot \ldots \cdot S^{b_\ell (n)}f_\ell }, \quad N\in {\mathbb {N}},\) 
we deduce that
\(\lim _{N\rightarrow \infty }{\mathbb {E}}_{n\in [N]}\, c_n \cdot S^{b_1(n)}f_1 \cdot \ldots \cdot S^{b_\ell (n)}f_\ell = 0\) 
in \(L^2(\mu )\) . This completes the proof.
